Measurement precision: typical error (two measurements of the same people)

In brief

If the same people were measured twice and nothing changed in between (test-retest), the typical error (TE) shows, in the unit of measurement, how far a single measurement is typically off. It appears as a collapsed addition next to Bland-Altman when exactly two number columns are chosen in the question "Do ratings agree?".

Requirements

• Two measurements of the same people, with no intervention, training or recovery in between. With two raters or two measurement methods the TE mixes their differences with the measurement error; Bland-Altman is the fitting reading then.
• At least 3 cases, the differences must not all be the same. The more people, the narrower the interval (rule of thumb: 20 or more).

Reading the result

• Typical error: the standard deviation of the differences divided by √2, with an interval via the χ² distribution (n − 1 degrees of freedom).
• Change of the mean: explicitly "second minus first measurement" (as Hopkins calculates it), with a t interval. Bland-Altman in Statable calculates first minus second measurement: the same change has the opposite sign there. Both labels are in the sheet.
• Coefficient of variation (CV, in percent): the typical error of the log-transformed values, back-transformed as 100 · (exp(TE) − 1). Only if all values are greater than 0; otherwise Statable says that it cannot be calculated. Statable does not show the approximation 100 · TE of the ln values because it is too small for larger errors.

Distinction

• The typical error is not the standard error of measurement (SEM) from the reliability (SEM = SD · √(1 − r)). Both share the idea, but the TE comes directly from the differences, without assuming a reliability.
• The reliable change index (RCI) asks whether a single person has changed reliably; it needs a reliability or retest numbers. The TE, in contrast, describes the precision of the measurement itself.
• The within-subject standard deviation (after Bland) includes the change of the mean; the TE does not.

Reporting the result

Typical error: TE = 0.48, 95% CI [0.36, 0.70]; change (second minus first measurement) = 0.43, 95% CI [0.12, 0.75]; CV = 3.7%, 95% CI [2.8, 5.4]

Example (20 sprint times in seconds, second measurement on average 3% slower): a single measurement typically deviates by 0.48 s. The typical error only goes into the report if "Show the typical error in the report" is ticked there.

For specialists

Own code (JavaScript) after Hopkins (2000): TE = SD(d)/√2 with SD from n − 1; interval TE · √(df/χ²0.975) to TE · √(df/χ²0.025) with df = n − 1; CV from the TE of the ln values with the same interval, then 100 · (exp(·) − 1). Checked against R (own formulas and the residual mean square of the analysis of variance y ~ person + time point).

Source

Hopkins, W. G. (2000). Measures of reliability in sports medicine and science. Sports Medicine, 30(1), 1-15. Bland, J. M. & Altman, D. G. (1996). Measurement error. BMJ, 312, 1654.